Spatial analysis for the distribution of cells in tissue sections

Spatial analysis, playing an essential role in data mining, is applied in a considerable number of fields. It is because of its broad applicability that dealing with the interdisciplinary issues is becoming more prevalent. It aims at exploring the underlying patterns of the data. In this project, we will employ the methodology that we utilize to tackle spatial problems to investigate how the cells distribute in the infected tissue sections and if there are clusters existing among the cells. The cells that are neighboring to the viruses are of interest. The data were provided by the Medetect Company in the form of 2-dimensional point data. We firstly adopted two common spatial analysis methods, clustering methods and proximity methods. In addition, a method for constructing a 2-dimensional hull was developed in order to delineate the compartments in tissue sections. A binomial test was conducted to evaluate the results. It is detectable that the clusters do exist among cells. The immune cells would accumulate around the viruses. We also found different patterns near and far away from viruses. This study implicates that the cells are interactive with each other and thus present the spatial patterns. However, our analyses are restricted in a planar circumstance instead of treating them in 3-dimensional space. For the further study, the spatial analysis could be carried out in three dimensions.It has been popular to utilize the heuristic methods or the existing methods to discover new findings and explain the mysterious phenomena in other subjects. And it is known that everything in nature relates to each other. In this sense, we could assume that the entire distribution of objects is relative to the locations of individuals. The idea of my work is attempting to explore this spatial relationship existing among cells. In my project, the relationships between individual cells or groups of cells are interesting. Our data is presented like the point cloud. It is doubted that if there are any groups existing among these points and if the viruses have neighbors. The methods are mainly categorized into three parts. The first method is to integrate the similar objects into groups. Here the similar objects could be the ones that are close to each other. The second method analyzes the degree of closeness between objects and looks for the neighbors of viruses. The last method can be used to draw the border of a point cloud, which seems like constructing the boundary of districts. Within each method, we carried out the corresponding case studies. Since similar objects can be grouped together, it is interesting to look into the details of each group. Thus we can know which two objects are similar in the same group. Basically, different types of cells in the same group can be checked and studied. In the closeness analysis, we found that some cells are indeed closer to each other. The constructed border could help us know the shape of point clouds. It can be concluded that the spatial relationship does exist among the cells. Groups of cells can be identified at a large extent. And one certain type of cells could be more attracted by some cells from a local level. However, this study is carried out in a 2D space. Actually, we neglect the real shape of cells which have heights. This could be a more interesting topic in the future

Mandoline: robust cut-cell generation for arbitrary triangle meshes

Although geometry arising "in the wild" most often comes in the form of a surface representation, a plethora of geometrical and physical applications require the construction of volumetric embeddings either of the geometry itself or the domain surrounding it. Cartesian cut-cell-based mesh generation provides an attractive solution in which volumetric elements are constructed from the intersection of the input surface geometry with a uniform or adaptive hexahedral grid. This choice, especially common in computational fluid dynamics, has the potential to efficiently generate accurate, surface-conforming cells; unfortunately, current solutions are often slow, fragile, or cannot handle many common topological situations. We therefore propose a novel, robust cut-cell construction technique for triangle surface meshes that explicitly computes the precise geometry of the intersection cells, even on meshes that are open or non-manifold. Its fundamental geometric primitive is the intersection of an arbitrary segment with an axis-aligned plane. Beginning from the set of intersection points between triangle mesh edges and grid planes, our bottom-up approach robustly determines cut-edges, cut-faces, and finally cut-cells, in a manner designed to guarantee topological correctness. We demonstrate its effectiveness and speed on a wide range of input meshes and grid resolutions, and make the code available as open source.This work is graciously supported by NSERC Discovery Grants (RGPIN-04360-2014 & RGPIN-2017-05524), NSERC Accelerator Grant (RGPAS-2017-507909), Connaught Fund (503114), and the Canada Research Chairs Program

An algorithm for the construction of intrinsic Delaunay triangulations with applications to digital geometry processing

The discrete Laplace–Beltrami operator plays a prominent role in many digital geometry processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate the use of the intrinsic Laplace–Beltrami operator. It satisfies a local maximum principle, guaranteeing, e.g., that no flipped triangles can occur in parameterizations. It also leads to better conditioned linear systems. The intrinsic Laplace–Beltrami operator is based on an intrinsic Delaunay triangulation of the surface. We detail an incremental algorithm to construct such triangulations together with an overlay structure which captures the relationship between the extrinsic and intrinsic triangulations. Using a variety of example meshes we demonstrate the numerical benefits of the intrinsic Laplace–Beltrami operator

An algorithm for the construction of intrinsic Delaunay triangulations with applications to digital geometry processing

The discrete Laplace-Beltrami operator plays a prominent role in many Digital Geometry Processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate use of the intrinsic Laplace-Beltrami operator. It satisfies a local maximum principle, guaranteeing, e.g., that no flipped triangles can occur in parameterizations. It also leads to better conditioned linear systems. The intrinsic Laplace-Beltrami operator is based on an intrinsic Delaunay triangulation of the surface. We give an incremental algorithm to construct such triangulations together with an overlay structure which captures the relationship between the extrinsic and intrinsic triangulations. Using a variety of example meshes we demonstrate the numerical benefits of the intrinsic Laplace-Beltrami operator